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arXiv:1109.3804 · Reviews in Mathematical Physics

QUANTUM HYPOTHESIS TESTING AND NON-EQUILIBRIUM STATISTICAL MECHANICS

V. JAKŠIĆ, Yoshiko Ogata, Claude‐Alain Pillet, Robert Seiringer

Published Jun 4, 2012• 73 citations
arXivHTMLPDF

Abstract

We extend the mathematical theory of quantum hypothesis testing to the general W*-algebraic setting and explore its relation with recent developments in non-equilibrium quantum statistical mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to quantum hypothesis testing of the arrow of time.

DOI
10.1142/s0129055x12300026

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We extend the mathematical theory of quantum hypothesis testing to the general W*-algebraic setting and explore its relation with recent developments in non-equilibrium quantum statistical mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to quantum hypothesis testing of the arrow of time.
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Hypothesis testing is an important task in mathematics and physics. Hypothesis testing of two random variables is related to the Kullback-Leibler divergence of the two corresponding distributions. Similarly, quantum hypothesis testing of two quantum states is characterized by the quantum relative entropy. While a quantum state can be abstracted as a device that only has outputs but no inputs, the most general quantum device is a quantum channel that also has inputs. In this work, we extend hypothesis testing to general quantum channels. In both the one-shot and asymptotic scenario, we study several quantifiers for hypothesis testing under different assumptions of how the channels are used. As the quantifiers are analogs to the quantum relative entropy of states, we call them the quantum relative entropy of channels. Then, we define the entropy of channels based on relative entropies from the target channel to the completely depolarizing channel. We investigate the properties that the quantum relative entropy of channels should satisfy, and we study its interplay with entanglement. With the broad applications of the quantum relative entropy of states, our results can be useful for understanding general properties of quantum channels.
In this paper, we study and answer the following fundamental problems concerning classical equilibrium statistical mechanics: 1): Is the principle of equal a priori probabilities indispensable for equilibrium statistical mechanics? 2): Is the ergodic hypothesis related to equilibrium statistical mechanics? Note that these problems are not yet answered, since there are several opinions for the formulation of equilibrium statistical mechanics. In order to answer the above questions, we first introduce measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. And we propose the measurement theoretical foundation of equili-brium statistical mechanics, and further, answer the above 1) and 2), that is, 1) is “No”, but, 2) is “Yes”.
We present our recent results on limit theorems for quantum entropies and their applications in quantum information theory. They include the full extensions of the fundamental asymptotic equipartition property (AEP) to ergodic quantum informations sources and the ergodic version of Stein's Lemma on quantum hypothesis testing. These limit theorems are then used to obtain a data compression theorem for ergodic quantum sources and a quantum version of Sanov's theorem on large deviations for uncorrelated quantum sources. Moreover, Stein's lemma clearifies the operational interpretation of the quantum Kullback-Leibler distance (relative entropy), and this operational understanding of this quantity can be employed to give a new intuitive proof of the monotonicity of this statistical distance measure relying on information-theoretic methods.